Fermi and von Neumann overlapped. They collaborated on problems of Taylor instabilities and they wrote a report. When Fermi went back to Chicago after that work he called in his very close collaborator, namely Herbert Anderson, a young Ph.D. student at Columbia, a collaboration that began from Fermi's very first days at Columbia and lasted up until the very last moment. Herb was an experimental physicist. (If you want to know about Fermi in great detail, you would do well to interview Herbert Anderson.) But, at any rate, when Fermi got back he called in Herb Anderson to his office and he said, "You know, Herb, how much faster I am in thinking than you are. That is how much faster von Neumann is compared to me."
There are a number of stories about von Neumann, but almost all of them are apocryphal. I do remember his former professors remarking that if they put unsolved problems on the chalkboard, he'd have them solved by the end of class.
Legend has it that when Gödel presented his incompleteness theorem Von Neumann understood that Hilbert's program was destroyed and immediately threw in the towel.
